1. No. Your desktop password manager is encrypted with a strong passphrase that locks when you’re computer locks. (Right?) They’d have to snatch your gear mid-session. Cookies are not safe, and cookie hijacks are a pretty common exploit. Cookies are for convenience, not security. Retaining authentication cookies is a very big security hole that we all do, and it’s why banks don’t let you re-auth on a previous session cookie.

2. “Pretty hard to break into” is the kind of phrase that keeps infosec people up at night.

3. Yes. First party cookies can be just as nefarious in addition to the technical requirements. 3a. Never assume that something supposed to be “mostly benign” isn’t currently being exploited for bad reasons.

To your OP, It’s actually not a terrible idea to uninstall the PW manager browser extension. It’s one more layer of isolation from the browser. You just lose the convenience of autofill.